Periodic oscillations arising and death in delay-coupled neural loops

We consider the system consisting of two identical n-neural loops which are coupled via a single neuron of each loop with discrete time delay. A framework describing the linearized stability region in the parameter space for the trivial steady state solution is constructed. In the boundary of the stability region, a periodic oscillation may arise from the Hopf bifurcation while these periodic oscillations could be dead owing to the change of coupling strength, delay or meeting a pitchfork bifurcation. The examples of n = 4 and n = 5 show the phenomena of oscillation arising or death, and the effects of the coupling strengths and delay on the stability and oscillation. The criticalities of pitchfork and Hopf bifurcations are investigated by using the normal form method for retarded functional differential equations.